MediaWiki 1.19 is a release that contains many new features and bug fixes.
This is a summary of the major changes of interest to users.
You can consult the RELEASE-NOTES file for the full list of changes in this version.
It was deployed to Wikimedia wikis in February 2012, see /Deployment announcement/ for a human-readable explanation.
Final version released on 2012-05-02 – you can download releases from the archive.
This is a Long Term Support release (LTS) which was supported until May 2015.

There are hundreds of bug fixes, new features, and changed features.
The comprehensive list of what's new is in the release notes.
Here's an overview:

Easier reading, editing, and maintenance

Disable the partial HTML and MathML rendering options for Math, and render as PNG by default (see RFC and r104498).MathML mode was so incomplete most people thought it simply didn't work.

The default user signature now contains a talk link in addition to the user link.

Better timezone recognition in user preferences.

Internationalization

The log rewrite has finally been completed, although some gender support features are still missing from this release (bug 24156). Documentation for developers on how to use the new logging system is (or will be) at Manual:Logging to Special:Log.

Heads-up for MediaWiki administrators

New skins/common/*.css files usable by skins instead of having to copy piles of generic styles from MonoBook or Vector's css.

BREAKING CHANGE: CSS styling for the wikitable class is now more specific to prevent it from affecting nested tables, which caused various issues (bug 30485 and bug 33434). If your wiki has overridden CSS rules for ".wikitable", please revise them and adjust where necessary. For comparison, use the "table.wikitable" section in skins/common/shared.css as base.

The maintenance script populateSha1.php was renamed to the more descriptive populateImageSha1.php.

There's a known issue where if PHP's PCRE is not compiled with the --enable-unicode-properties all text information on the installer setup page will be replaced with several instances of [1]. MediaWiki 1.19 requires the PCRE library used with PHP to be compiled with --enable-unicode-properties in order to function.

And more!

ResourceLoader

ResourceLoaderWikiModule are now also automatically purged if one of the messages used from the MediaWiki namespace are updated.

ResourceLoader's loading system now supports custom sources for loading the modules (see $wgResourceLoaderSources for more information)

"private" ResourceLoader modules embed has been moved from towards the </body> end, to up in the <head>. The embedding can no longer be disabled ($wgResourceLoaderInlinePrivateModules removed)

Popular ResourceLoader modules (e.g. mediawiki.util) are now preloaded as stop-gap for scripts missing dependencies.

New configuration variable $wgPreloadJavaScriptMwUtil has been introduced for this (set to false by default for new installations). Set to true if your wiki has a large amount of user/site scripts that are lacking dependency information. In the short to medium term these user/site scripts should be fixed by adding the used modules to the dependencies in the module registry and/or wrapping them in a callback to mw.loader.using.

Watchlist

Special:Watchlist no longer sets links to feed when the user is anonymous.

Watchlist editing has been improved so that users can now remove broken or invalid titles.

wlexcludeuser parameter added to ApiFeedWatchlist.

Action=watch now parses messages using the correct title instead of "API".

Logging

Searching blocked usernames in block log is now clearer. The "User:" prefix is no longer necessary when searching for a user's block log entry.

Log comments are now truncated correctly in multibyte character languages.